Extreme weights in Steinhaus triangles

Let {0=w0<w1<w2<…<wm0=w0<w1<w2<…<wm} be the set of weights of binary Steinhaus triangles of size n , and let Wibe the set of sequences in F2n that generate triangles of weight wi. In this paper we obtain the values of wi and the corresponding sets Wi for i¿{2,3,m}i¿{2,3,m}, and partial results for i=m-1i=m-1.Peer ReviewedPostprint (author's final draft

On the problem of Molluzzo for the modulus 4

We solve the currently smallest open case in the 1976 problem of Molluzzo on $\mathbb{Z}/m\mathbb{Z}$, namely the case $m=4$. This amounts to constructing, for all positive integer $n$ congruent to $0$ or $7 \bmod{8}$, a sequence of integers modulo $4$ of length $n$ generating, by Pascal's rule, a Steinhaus triangle containing $0,1,2,3$ with equal multiplicities.Comment: 12 pages ; 3 figures ; 3 tables, Integers : Electronic Journal of Combinatorial Number Theory, State University of West Georgia, Charles University, and DIMATIA, 2012, 12, pp.A1

A universal sequence of integers generating balanced Steinhaus figures modulo an odd number

In this paper, we partially solve an open problem, due to J.C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer $n$, that are Steinhaus triangles containing all the elements of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. For every odd number $n$, we build an orbit in $\mathbb{Z}/n\mathbb{Z}$, by the linear cellular automaton generating the Pascal triangle modulo $n$, which contains infinitely many balanced Steinhaus triangles. This orbit, in $\mathbb{Z}/n\mathbb{Z}$, is obtained from an integer sequence called the universal sequence. We show that there exist balanced Steinhaus triangles for at least $2/3$ of the admissible sizes, in the case where $n$ is an odd prime power. Other balanced Steinhaus figures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascal trapezoids or lozenges, also appear in the orbit of the universal sequence modulo $n$ odd. We prove the existence of balanced generalized Pascal triangles for at least $2/3$ of the admissible sizes, in the case where $n$ is an odd prime power, and the existence of balanced lozenges for all admissible sizes, in the case where $n$ is a square-free odd number.Comment: 30 pages ; 10 figure

Regular Steinhaus graphs of odd degree

A Steinhaus matrix is a binary square matrix of size $n$ which is symmetric, with diagonal of zeros, and whose upper-triangular coefficients satisfy $a_{i,j}=a_{i-1,j-1}+a_{i-1,j}$ for all $2\leq i<j\leq n$. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states that even Steinhaus graphs, i.e. those with all vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek conjectured that the complete graph on two vertices $K_2$ is the only regular Steinhaus graph of odd degree. Using Dymacek's theorem, we prove that if $(a_{i,j})_{1\leq i,j\leq n}$ is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree then its sub-matrix $(a_{i,j})_{2\leq i,j\leq n-1}$ is a multi-symmetric matrix, that is a doubly-symmetric matrix where each row of its upper-triangular part is a symmetric sequence. We prove that the multi-symmetric Steinhaus matrices of size $n$ whose Steinhaus graphs are regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend on $\lceil \frac{n}{24}\rceil$ parameters for all even numbers $n$, and on $\lceil \frac{n}{30}\rceil$ parameters in the odd case. This result permits us to verify the Dymacek's conjecture up to 1500 vertices in the odd case.Comment: 16 page

Balanced simplices

An additive cellular automaton is a linear map on the set of infinite multidimensional arrays of elements in a finite cyclic group $\mathbb{Z}/m\mathbb{Z}$. In this paper, we consider simplices appearing in the orbits generated from arithmetic arrays by additive cellular automata. We prove that they are a source of balanced simplices, that are simplices containing all the elements of $\mathbb{Z}/m\mathbb{Z}$ with the same multiplicity. For any additive cellular automaton of dimension $1$ or higher, the existence of infinitely many balanced simplices of $\mathbb{Z}/m\mathbb{Z}$ appearing in such orbits is shown, and this, for an infinite number of values $m$. The special case of the Pascal cellular automata, the cellular automata generating the Pascal simplices, that are a generalization of the Pascal triangle into arbitrary dimension, is studied in detail.Comment: 33 pages ; 11 figures ; 1 tabl

Rotational and dihedral symmetries in Steinhaus and Pascal binary triangles

We give explicit formulae for obtaining the binary sequences which produce Steinhaus triangles and generalized Pascal triangles with rotational and dihedral symmetries.Postprint (published version

On a problem of Molluzzo concerning Steinhaus triangles in finite cyclic groups

Let $X$ be a finite sequence of length $m\geq 1$ in $\mathbb{Z}/n\mathbb{Z}$. The \textit{derived sequence} $\partial X$ of $X$ is the sequence of length $m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of iterated derived sequences of $X$, until length 1 is reached, determines a triangle, the \textit{Steinhaus triangle $\Delta X$ generated by the sequence $X$}. We say that $X$ is \textit{balanced} if its Steinhaus triangle $\Delta X$ contains each element of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. An obvious necessary condition for $m$ to be the length of a balanced sequence in $\mathbb{Z}/n\mathbb{Z}$ is that $n$ divides the binomial coefficient $\binom{m+1}{2}$. It is an open problem to determine whether this condition on $m$ is also sufficient. This problem was posed by Hugo Steinhaus in 1963 for $n=2$ and generalized by John C. Molluzzo in 1976 for $n\geq3$. So far, only the case $n=2$ has been solved, by Heiko Harborth in 1972. In this paper, we answer positively Molluzzo's problem in the case $n=3^k$ for all $k\geq1$. Moreover, for every odd integer $n\geq3$, we construct infinitely many balanced sequences in $\mathbb{Z}/n\mathbb{Z}$. This is achieved by analysing the Steinhaus triangles generated by arithmetic progressions. In contrast, for any $n$ even with $n\geq4$, it is not known whether there exist infinitely many balanced sequences in $\mathbb{Z}/n\mathbb{Z}$. As for arithmetic progressions, still for $n$ even, we show that they are never balanced, except for exactly 8 cases occurring at $n=2$ and $n=6$.Comment: 29 pages, 10 figure